Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework

• Published in: Journal of scientific computing Vol. 55; no. 3; pp. 718 - 737
• Main Authors: Chen, Yanlai, Gottlieb, Sigal
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-06-2013; Springer Nature B.V
• Subjects: Algorithms; Approximation; Collocation; Collocation methods; Computational Mathematics and Numerical Analysis; Cost engineering; Empirical analysis; Galerkin method; Galerkin methods; Least squares method; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; On-line systems; Partial differential equations; Strategy; Theoretical; Greedy algorithms; Collocation method; Reduced collocation method; Reduced basis method; Least squares
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-012-9654-z

In this paper, we present
the first
reduced basis method well-suited for the collocation framework. Two fundamentally different algorithms are presented: the so-called Least Squares Reduced Collocation Method (LSRCM) and Empirical Reduced Collocation Method (ERCM). This work provides a reduced basis strategy to practitioners who prefer a collocation, rather than Galerkin, approach. Furthermore, the empirical reduced collocation method
eliminates
a potentially costly online procedure that is needed for non-affine problems with Galerkin approach. Numerical results demonstrate the high efficiency and accuracy of the reduced collocation methods, which match or exceed that of the traditional reduced basis method in the Galerkin framework.